There is a peculiar kind of proof in mathematics that tells you something exists without telling you what it looks like, where to find it, or how to build it. It simply shows you, with iron-clad logic, that it must be out there somewhere. This type of proof strikes many people as almost philosophically unsatisfying, how can you prove something exists without actually producing it? And yet it is one of the most powerful ideas in all of modern mathematics.
That idea has a name: the probabilistic method. It was invented in 1947 by Paul Erdős, the legendary itinerant Hungarian mathematician who spent his life wandering from university to university, collaborating with hundreds of mathematicians across the world, and leaving behind a trail of problems, conjectures and breakthroughs that kept the mathematics community busy for generations. The probabilistic method was perhaps his greatest gift to the field. And for nearly 80 years, no one had found a way to significantly improve on it for the class of problems Erdős originally cared about most.
Until a graduate student at Tsinghua University decided to try.
The Problem: Colouring Connections
To understand what makes this story remarkable you first need to understand the problem Erdős was trying to solve. Imagine a network of nodes – points – where every pair of nodes is connected by a line. Now colour each line either red or blue. Your challenge is to avoid creating large clusters where every member is connected to every other member by lines of the same colour. These forbidden clusters are called monochromatic cliques.
It sounds like a puzzle. And in a sense it is. But as the network gets bigger, the puzzle becomes impossible. At some point, no matter how cleverly you colour the lines, a monochromatic clique of a certain size will inevitably appear somewhere in the network. The question mathematicians call a Ramsey number asks: how big can the network get before a monochromatic clique of a given size is unavoidable?
The Ramsey number for a clique of size three, for instance, is six. If you have six nodes all connected to each other, you cannot colour the lines red and blue without creating a triangle of the same colour somewhere. Five nodes you can manage. Six you cannot.
As the clique size grows the problem becomes extraordinarily difficult. Mathematicians have been able to calculate only a handful of the smallest Ramsey numbers. As Paul Horn of the University of Denver put it: “It’s very hard to create something that has no structure. Maybe it’s because we’re human and we’re subject to our biases.”
Erdős and the Power of Randomness
This is where Erdős made his revolution. Rather than trying to build a clique-free network by hand which is fiendishly difficult, he asked a different question. What if you consider every possible way to colour the network and choose one at random? He then showed that the probability of getting a clique-free colouring is greater than zero. Which means such a colouring must exist, even if you have no idea what it looks like.
Before the development of the probabilistic method, “if I’m telling you that certain objects exist, you would tell me, ‘Show me,'” said Benny Sudakov, a mathematician at the Swiss Federal Institute of Technology Zurich. “But certain objects are so unusual that it’s hard for us to grasp that they exist at all.”
The proof was just a few lines long. But its impact was enormous. At first, mathematicians were loath to follow Erdős’ lead. They wanted concrete examples. “For many years, Erdős was like a voice in the wilderness,” said Joel Spencer of New York University. “He was getting these amazing results using randomness, and people had never done that before.”
But soon the method proved its worth beyond doubt. Today it is used across mathematics and computer science, to test whether a number is prime, to design better circuits, to clean up data without introducing biases. As Spencer put it: “Now, that’s the baseline.”
Yet for all its power, the probabilistic method had a stubborn limitation. For the specific class of Ramsey numbers Erdős originally cared about, what mathematicians call near-diagonal Ramsey numbers, where the forbidden red and blue cliques are similar in size, progress had essentially frozen. Erdős showed in 1947 that the Ramsey number for same-size cliques of size k must be bigger than roughly the square root of 2 raised to the power k. Over the following eight decades, the entire combined effort of the global mathematics community managed to nudge that bound from 2 to the power of 500 all the way to 2 to the power of 501. For half a century on the near-diagonal problem, the needle barely moved.
Enter a Graduate Student With an Unusual Idea
Wujie Shen had spent his first few semesters at Tsinghua University focused mainly on geometry and topology. But in the spring of 2024, he came across a paper on Ramsey numbers that captivated him.
Shen understood how Erdős’ method worked. You flip a coin to decide the colour of each connection in the network, heads for red, tails for blue. You then calculate the probability that this random colouring will be clique-free. But this calculation becomes extraordinarily difficult for large networks. Shen wondered whether there was a smarter way to introduce randomness, a model that could produce clique-free colourings more efficiently than Erdős’ simple coin flip.
Given his background in geometry, the model he came up with was unexpected. He wanted to use the geometry of high-dimensional spheres.
The Strange World of High-Dimensional Spheres
A sphere, in everyday life, is something you can hold in your hand, a round shape, every point on its surface equidistant from the centre. But mathematicians can define spheres in any number of dimensions, and the higher the dimension the stranger the behaviour. These spheres “mess with all our intuitions completely,” said David Conlon of the California Institute of Technology. Many of our assumptions about what a sphere looks like are no longer true in high dimensions: a high-dimensional sphere has a tiny volume and massive surface area, and most of its points lie on the equator. It’s “pretty complicated to work with,” Sudakov said.
Shen, working with his colleague Jie Ma and Ma’s graduate student Shengjie Xie, proposed a new colouring method. First, place all the nodes of your network randomly on the surface of a high-dimensional sphere. Then colour each connection based on the distance between the two nodes it connects, if the nodes are far apart, colour the connection red; if they are close together, colour it blue.
The insight behind this approach is elegant. To form a large red clique, you need many nodes that are all far from each other. But on the surface of a sphere, with only so much space available, many nodes being simultaneously far from each other is geometrically constrained. The geometry works against the formation of large red cliques in a way that Erdős’ coin flip method does not exploit.
There was a catch, however. By the same logic, the method also produced more blue cliques than Erdős’ approach. “There’s a trade-off that looks like it really helps in one colour, but it doesn’t help at all in the other colour,” Conlon said. “Why bother?”
But Ma, Shen and Xie persisted. They tested their approach on smaller networks and found that even among the many bad colourings their method produced, there was still a nonzero chance of getting a good clique-free colouring. The benefits, they believed, could outweigh the costs even for much larger networks.
The Proof and Its Breakthrough
The key to unlocking the geometry turned out to lie in a peculiar property of high-dimensional spheres. When you randomly place many nodes on the surface of such a sphere and draw lines from each node to the centre, those lines are almost always perpendicular to each other. This is not true in two or three dimensions, it is a feature that only emerges in very high dimensions. And this near-universal perpendicularity restricts how far apart the nodes can be from each other, which in turn limits their chances of forming a large monochromatic clique.
After a year and 40 pages of dense computations, the trio published their paper in July 2025. The result was the first improvement for near-diagonal Ramsey numbers in 50 years. The change in the growth rate was tiny, they managed to nudge Erdős’ growth rate up by a factor of 10 to the power of minus 21, but the significance is enormous. A 50-year frozen problem had moved.
“It’s lucky, and we feel like all our efforts are rewarded,” Ma said. “But it was tough for a long time.”
“It’s a bit shocking that a familiar thing works for a familiar problem,” said Julian Sahasrabudhe of the University of Cambridge. Their technique, he said, “was hidden in plain view.”
What Happened Next
Mathematics rarely stands still after a breakthrough like this. In December 2025, Sudakov and two of his graduate students drastically simplified the team’s colouring model, improving their new bounds even further. Others have since used the model to estimate Ramsey numbers that involve three colours, not two.
This ripple effect is characteristic of the probabilistic method’s entire history. For 80 years mathematicians have been tinkering with Erdős’ randomness-based technique, finding new ways to mix in additional structure to boost its power. Each improvement has then proved useful far beyond the original problem.
Why This Story Matters
For students who encounter mathematics through school, competitions or casual curiosity, this story carries several lessons worth holding onto.
The first is about the power of an unexpected idea. Erdős’ insight, that you could prove something exists by showing it cannot be impossible, was so strange when he proposed it that the mathematical community took years to accept it. Today it underpins huge swathes of modern mathematics and computer science.
The second is about the value of crossing boundaries. Wujie Shen was a graduate student trained in geometry who wandered into a problem in graph theory. His outsider perspective was precisely what unlocked a method that specialists had been staring at for decades. As Sahasrabudhe noted, the technique was hidden in plain view. Sometimes the person who sees it is the one who was not looking in the expected direction.
The third is about the nature of progress in mathematics. The improvement Ma, Shen and Xie achieved was tiny in numerical terms. A change of 10 to the power of minus 21 sounds almost negligible. But in mathematics what matters is not the size of the step — it is whether the barrier has been crossed. For 50 years the near-diagonal Ramsey problem had not moved at all. Now it has. And once a barrier moves, it tends to keep moving.
Erdős himself spent his life believing that mathematics was a vast and largely unexplored territory, full of beautiful truths waiting to be discovered. He carried almost nothing with him as he wandered the world, a single suitcase, a few changes of clothes, and an inexhaustible supply of problems. He would have been delighted to know that 80 years after his greatest invention, a graduate student on the other side of the world had found a way to make it stronger.
That, perhaps, is the most enduring lesson of all. Mathematical ideas do not expire. They wait.
